The fallacy in Locke's argument for the existence of God (or at least one fallacy) has been noted by readers of the Essay since Leibniz. The argument starts with two premises that are unassailable, that "there is some real Being" and that "Non-entity cannot produce any real Being". From these two premises, Locke concludes (validly, if the second premise is properly interpreted)that (A) "From eternity there has been something" (Essay IV.x.3). Then in the next paragraph he refers to "This eternal Source . . . of all being", as if the argument so far entitles him to hold that (B) one single being has existed from eternity. Perhaps he thinks that (B) is merely a restatement of (A), that the two have the same meaning; or perhaps he thinks that (B) follows from (A). If the former, Locke's fallacy is that of equivocation, since (B) and (A) do not have the same meaning; if the latter, then he is simply mistaken about the logical relation between (A) and (B). (Contemporary logicians would say that in either case, Locke gets from (A) to (B) by switching the order of the two quanitifiers that are implicit in both).
Except that Locke commits neither fallacy as stated. You can see the original argument here. The mistake is in thinking that Locke moves directly from (A) to (B). But this is clearly not the case. The idea is:
(1) Everything not from eternity has a beginning;
(2) Everything that has a beginning must be produced by something else;
(3) Therefore there must be something from eternity.
Obviously, there are implicit premises here that would have to be investigated. But there is no fallacy of equivocation (except in the misinterpretations of Locke's critics), and no misdiagnosis of the logical structure of the argument (except on the part of Locke's critics). Locke's critics are confused about the role played by paragraph 3 in the argument: paragraph 3 is introductory; the details about the 'something' that paragraph 3 argues for are worked out in the next few paragraphs. Chappell's mention of the next paragraph is itself a bit of illogical gerrymandering: the phrase he mentions is the conclusion of a further argument about the power that must be attributed to the something (it must be a unified cause), not Locke's summary of the conclusion of the paragraph 3 argument. There is no doubt that Locke's arguments aren't in rigorous form; there's also no doubt that blatant misreadings of Locke's arguments shouldn't be held against him. If Chappell, to make the fallacy charge stick, has to radically rearrange the order in which Locke talks about things, that's a sign that perhaps Locke was right to tackle them in the order he actually did.
UPDATE: And what gets me about this charge is that it is so obviously false. Locke's argument is clearly constructed in stages, which can be organized by their conclusions:
1) Something must exist from eternity (i.e., there must be something that did not begin from another). (IV.x.3)
2) Something that exists from eternity must be the unified source for all things that do begin from another. (IV.x.4)
3) Something that exists from eternity as a unified source must have knowledge. (IV.x.5)
4) God (as eternal, most powerful, most knowledgeable being) must exists. (IV.x.6)
You can read it yourself and see that this is exactly how Locke himself sets it up. And it makes sense to do it this way, because given (4) someone might argue (a) that there is nothing eternal; (b) that what is eternal is not a unified source of things that are not eternal; or (c) that this eternal, unified source does not have knowledge. Locke's argument is not rigorously constructed, and could only be made so by a generous use of implicit premises, and it's not clear that all his arguments are particularly sound anyway, but Locke is so obviously not guilty of the fallacies of which he is accused that it is astonishing to me the claim is even made.
[A commenter claims I make the mistake Locke is accused of in (3) and (1). I can see how the confusion would arise -- it comes from taking 'something' here to mean 'something in particular'; but it isn't intended to mean that either here or in Locke's discussion, but the more general 'something (whatever that something may be)'. In any case, it is still clear that only by arbitrarily rearranging Locke's argument can you get the result Chappell claims.]